Topics in Quasiconformal mappings and in PDE

在这个研究项目中研究的问题有一个共同的主题，即从几何和梯度流中产生的泛函的临界点的定性研究。更具体地说，首席研究员将继续研究弱解的非线性偏微分方程（PDE）的家庭是次黎曼模拟的平均曲率流PDE，最小曲面PDE，和共形的n-Laplacian。 首席研究员还将研究几何函数理论/非线性PDE方法，以解决强伪凸集之间的双全纯映射的边界正则性，并继续研究空间中的极值拟共形映射。 在自然科学和工程中，研究复杂系统的平衡态是很重要的。数学允许模拟这些状态作为能量泛函的临界点，并通过分析与这些泛函相关的某些微分方程来研究它们的性质。这些方程的解的性质取决于“背景几何”，该“背景几何”对诸如材料的非均匀性或约束的存在（诸如在机器人臂的运动中等）等现实特征进行建模。这种“背景几何学”最普遍的例子之一是亚黎曼几何学，它对运动只能沿着沿着给定方向的空间进行建模，如在复杂分析、机器人手臂运动和控制理论、量子计算、卫星、量子力学以及哺乳动物视觉皮层的结构功能中的应用。PI将与他的学生以及他的合作者一起开展这项工作。 亚黎曼几何的研究从其起源起就受到“真实的世界”问题和PI讲座的推动，从研究生水平到高中水平的各种观众。

The problems studied in this research project have as common theme the qualitative study of critical points of functionals arising from geometry and of their gradient flows. More specifically the principal investigator will continue to study weak solutions to a family of nonlinear partial differential equations (PDE) that are the sub-Riemannian analogues of the mean curvature flow PDE, the minimal surface PDE, and the conformal n-Laplacian. The principal investigator will also study a geometric function theory/nonlinear PDE approach to the boundary regularity of biholomorphic mappings between strongly pseudoconvex sets and pursue the study of extremal quasiconformal mappings in space. Both in the natural sciences and in engineering it is important to study equilibrium states of complex systems. Mathematics allows to model such states as critical points of energy functionals and to study their properties by analyzing certain differential equations associated to these functionals. The properties of the solutions to these equations depend on a 'background geometry' that models such real-life features as the non-homogeneity of materials, or the presence of constraints (such as in the motion of robot arms, etc. etc. ...). One of the most ubiquitous instances of such 'background geometry' is sub-Riemannian geometry, modeling spaces where motion is possible only along a given set of directions, as in applications to complex analysis, motion of robot arms and control theory, quantum computing, satellites, quantum mechanics and in the structural functions of the mammalian visual cortex. The PI will pursue this work together with his students as well as his collaborators. The study of sub-Riemannian geometry has been driven since its origins by "real world" problems and the PI lectures to a wide variety of audiences from the post-graduate level to the high school level.